Insurance premiums should not rise as a result of HCD designation. Designation itself does not place additional requirements on the insurer and should not negatively affect premiums. For instance, there is no requirement that repairs be made with antique material.
A variety of other factors can cause insurance companies to increase premiums for older buildings, such as a higher level of risk due to out-dated wiring that has not been upgraded or old heating systems, etc.
You should consult your insurance broker for details.